You're missing the third person plural (presumably male) pronoun. The conjunction " 'anna " takes the subject of the coming clause in the accusative immediately after it. Here, you would affix "hum" to it.

Isn’t Arabic like Spanish in that it does not require a subject. For example, can’t I say اكل or do I need to say انا اكل?

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That's right. But, 'anna is from a group of conjunctive particles that requires the subject of the coming clause to immediately follow in the accusative. You don't have an option. If the subject is a pronoun, that enclitic pronoun is affixed to the conjunctive itself.

The only exception is if the subject is indefinite and the predicate is a prepositional phrase. Then, the prepositional phrase comes first and the subject follows.

'anna is from a group of conjunctive particles that requires the subject of the coming clause to immediately follow in the accusative. You don't have an option. If the subject is a pronoun, that enclitic pronoun is affixed to the conjunctive itself.

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If أنَّ must be followed by the pronoun corresponding to the subject of the clause following it, then shouldn’t the phrase:
هل تستطيعين أن تترجمي هذا المقطع؟
instead be:
هل تستطيعين أنِك تترجمي هذا المقطع؟

If أنَّ must be followed by the pronoun corresponding to the subject of the clause following it, then shouldn’t the phrase:
هل تستطيعين أن تترجمي هذا المقطع؟
instead be:
هل تستطيعين أنِك تترجمي هذا المقطع؟

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'an is not in the set of verb like conjunctives that 'anna is in. So, the second sentence is incorrect.

'an and 'anna are both translated as "that", but 'an is used differently. It evokes the subjunctive after it, while 'anna does not.

The first one is correct.
The second is not correct because you mention both the noun الأولاد and the pronoun refering to it ـهم one next to the other and the pronoun preceding the noun to which it refers. It's like saying in English: I think that theythe boys sang.